Matrix eigenvalue problem

Another step that is common to most, if not all, approaches that compute orbitals of one fomi or anotiier is tlie solution of matrix eigenvalue problems of the fomi... 

This is because no four-indexed two-electron integral like expressions enter into the integrals needed to compute the energy. All such integrals involve p(r) or the product p(/)p(r) because p is itself expanded in a basis (say of M functions), even the term p(r)p(r) scales no worse than tvF. The solution of the KS equations for the KS orbitals ([). involves solving a matrix eigenvalue problem this... 

The two sets of coeflicien ts, one for spin-up alpha electrons and the other for spin-down beta electrons, are solutions of iw O coupled matrix eigenvalue problems ... 

This matrix eigenvalue problem then beeomes ... 

As in all matrix eigenvalue problems, we are able to express (n-1) elements of the eigenveetor v(k) in terms of one remaining element. However, we ean never solve for this one last element. So, for eonvenienee, we impose one more eonstraint (equation to be... 

Remember that ai is the representation of g(x) in the fi basis. So the operator eigenvalue equation is equivalent to the matrix eigenvalue problem if the functions fi form a complete set. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

The LCAO expansion coefficients are found from the matrix eigenvalue problem... 

The next step might be to perform a configuration interaction calculation, in order to get a more accurate representation of the excited states. We touched on this for dihydrogen in an earlier chapter. To do this, we take linear combinations of the 10 states given above, and solve a 10 x 10 matrix eigenvalue problem to find the expansion coefficients. The diagonal elements of the Hamiltonian matrix are given above (equation 8.7), and it turns out that there is a simplification. 

This is clearly a matrix eigenvalue problem the eigenvalues determine tJie vibrational frequencies and the eigenvectors are the normal modes of vibration. Typical output is shown in Figure 14.10, with the mass-weighted normal coordinates expressed as Unear combinations of mass-weighted Cartesian displacements making up the bottom six Unes. 

The determinants Jj form now a very convenient set of trial functions. The Raleigh-Ritz variational principle, keeping the determineuits As fixed and varying only the coefficients aim -. -nipin, leads to the following matrix eigenvalue problem ... 

Setting up the matrix eigenvalue problem Eq. (18) for the five determinants gives ... 

The numerical solution of large matrix eigenvalue problems is one of the most frequently encountered computation tasks of a standard, yet nontrivial nature. Consequently, there is great theoretical and practical interest in the invention, modification and assessment of methods to cope with this problem. This chapter will describe some well-established methods, currently of widespread use in quantum-chemistry programs. 

This is not so for the matrix eigenvalue problem the eigenvalues (and eigenvectors) of real matrices n generally can only be found in the complex plane C (and in C"). The... 

The poles of the frequency-dependent polarizability correspond to electronic excitations, occurring with an infinitesimal perturbation, i.e., h = 0. Substituting this into Eq. (2-17) leads to a nonsymmetric matrix eigenvalue problem ... 

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